VERTICAL-CAVITY Surface-Emitting Lasers have been

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 23, DECEMBER 1, 2016 5437 Polarization- and Modal-Control in a Vertical-Cavity Surface-Emitting Laser With an External-Cavity Formed by a Liquid Crystal Overlay Leszek Frasunkiewicz, Tomasz Czyszanowski, Hugo Thienpont, and Krassimir Panajotov Abstract By utilizing a fully-vectorial three-dimensional coldcavity optical simulation, we analyze the spectral, polarization, and modal-losses behavior of a vertical-cavity surface-emitting laser with a nematic liquid-crystal (LC) overlay. We identify phenomena responsible for mode and polarization selectivities and show how they can be controlled by varying design geometry parameters, such as LC thickness and oxide aperture diameter, or by electrooptically tuning the LC director. We present possibility of achieving voltage-controlled polarization switching of a fundamental mode or transverse mode switching within the same polarization. Furthermore, we show that it is theoretically possible to obtain improved single-mode operation by means of increased higher order transverse mode discrimination. Finally, we point out a way to destabilize the diode by equalizing the modal losses of all the analyzed modes. Index Terms Coupled-cavities, mode switching, nematic liquid crystal, numerical analysis, optical simulations, polarization control, vertical-cavity surface-emitting laser. I. INTRODUCTION VERTICAL-CAVITY Surface-Emitting Lasers have been valued sources of light for applications such as short haul optical telecommunication, spectroscopy, or computer mice [1]. Low threshold current, low power consumption, high conversion efficiency, cylindrical geometry, single fundamental mode operation and ease of mass manufacturing makes them a Manuscript received May 31, 2016; revised August 2, 2016 and September 15, 2016; accepted October 13, 2016. Date of publication October 18, 2016; date of current version November 7, 2016. The work of K. Panajotov was supported in part by the Interuniversity Attraction Poles program of the Belgian Science Policy Office under Grant IAP P7-35, in part by the Methusalem foundation, and in part by the Project DFNI-T02/18 of Bulgarian National Science Fund. The work of H. Thienpont was supported in part by the Interuniversity Attraction Poles program of the Belgian Science Policy Office under Grant IAP P7-35 and in part by the Methusalem foundation. The work of L. Frasunkiewicz and T. Czyszanowski was supported under Project DEC-2012/06/M/ST7/00442 from the Polish National Science Centre. L. Frasunkiewicz is with the Faculty of Engineering Sciences, Brussels Photonics Team B-PHOT, Vrije Universiteit Brussel, Brussels 1050, Belgium, and also with the Institute of Physics, Technical University of Lodz, Lodz 90-924, Poland (e-mail: lfrasunk@b-phot.org). T. Czyszanowski is with the Institute of Physics, Technical University of Lodz, Lodz 90-924, Poland (e-mail: tomasz.czyszanowski@p.lodz.pl). H. Thienpont is with the Faculty of Engineering Sciences, Brussels Photonics Team B-PHOT, Vrije Universiteit Brussel, Brussels 1050, Belgium (e-mail: hthienpo@b-phot.org). K. Panajotov is with the Faculty of Engineering Sciences, Brussels Photonics Team B-PHOT, Vrije Universiteit Brussel, Brussels 1050, Belgium, and also with the Institute of Solid State Physics, Sofia 1784, Bulgaria (e-mail: kpanajot@b-phot.org). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2016.2619178 device of choice for high speed parallel optical links in supercomputers and as computer mice sensors. Tolerance to broad ambient temperature conditions and possibility of wavelength tuning are additional properties important in gas-sensing. On the other hand, the cylindrical symmetry of VCSELs results in a lack of incorporated polarization-selective mechanisms [1], [2] and switching between two orthogonal polarizations of linearly polarized laser modes often occurs during VCSEL operation [1], [3], [4], [5]. Such behavior is strongly depreciated in many applications, including the aforementioned areas. Various approaches for polarization control in VCSELs have been investigated [1, Ch. 5], with shallow surface gratings etched in the topmost Distributed Bragg Reflector (DBR) layer being the most successful and commercially widespread solution [6], [7]. However, such approach is basically introducing a polarization-discrimination mechanism in the laser and does not provide flexibility in controlling the polarization-state. This additional degree of freedom can be achieved by means of optical injection [8], [9], [10], [11] or optical-feedback [12], [13]. Examples of such approach include a Liquid Crystal induced polarization-sensitive feedback [14] or extremely short externalcavity [13]. Unfortunately, the design proposed in [14] is rather bulky and impractical, while the short air-filled cavities provide a very limited polarization selection. However, it was shown theoretically [15] and experimentally [16] that these issues can be addressed by incorporating LC in a short external VCSEL cavity, which results in improved polarization-selectivity and reduced complexity of the design. Manufacturing process of such structure could be even more simplified by utilizing a two-wafer bonding method [17]. While single-mode polarization characteristics in an externalcavity LC-VCSEL were investigated in detail, no analysis of higher-order transverse modes and their polarization behavior has been carried out so far. Since standard oxide-confined VC- SELs usually emit several transverse modes, it is important to verify characteristics of these higher-order modes of both polarizations, to see how they interact with the fundamental mode. Numerical analysis of such phenomena requires a fullyvectorial, three-dimensional optical model. For this reason a Plane Wave Admittance Method (PWAM) [18] will be utilized, which has already been successfully used to simulate complex VCSEL structures, such as Photonic-Crystal VCSELs [19], revealing very good agreement with experiment [20]. The PWAM is relatively fast computational method (with respect to other vectorial methods) of high accuracy, because it combines analytical description of light propagation in the z-direction, 0733-8724 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

5438 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 23, DECEMBER 1, 2016 Fig. 1. (color online) Schematic drawing of the LC-VCSEL. L is the thickness of the LC film in the optical axis of the device between the VCSEL diode and the golden mirror (LC cavity), r ox is the VCSEL oxide-aperture radius and θ, shown in in the inset, is the average angle between the LC molecules and the Oz axis in the yoz plane (LC director). The small red rectangles schematically indicate the presence of LC molecules, oriented after the LC director in the LC cavity, when there is no electric field applied. perpendicular to the multilayered VCSEL structure, with plane-wave decomposition in the transverse x-y planes of each VCSEL layer [18]. In this paper we investigate in detail the interplay between fundamental-mode polarization switching and transverse-mode switching, induced by voltage tuning of LC director orientation in 850 nm LC-VCSEL. We demonstrate that by a careful choice of the LC thickness one can employ the LC-VCSEL to perform various voltage-controlled functions. It is possible to obtain either fundamental-mode polarization switching and simultaneous increased discrimination of higher-order modes, or transverse-mode switching with simultaneous strong polarization discrimination, or increased laser instability induced by competitions of polarization and transverse modes. The paper is organized as follows. In Section II a description of the simulated structure will be given, followed by an analysis of the transverse modes and information about the numerical procedure in Section III. In Section IV the impact of the various geometry parameters on the spectral, modal and polarization characteristics of the LC-VCSEL will be presented, which will allow us to choose several representative cases for describing voltage controlled polarization-switching in Section V. Finally, the work will be concluded in Section VI. II. LC-VCSEL STRUCTURE The structure considered in this paper (Fig. 1) is based on [15], [16] and is an 850 nm emitter realized using GaAs/AlGaAs materials with an aluminum oxide layer for transverse optical confinement. LC-VCSELs can be manufactured in different ways. In the approach of [16] the VCSEL chip is incorporated in a LC cell, i.e. between two glass plates covered with transparent contact (ITO). The top ITO layer is patterned by using lithography in order to define separate contacts for the VCSEL anode and for the LC cell. A thin photo-alignment layer is spincoated on the cathode and anode, which are then electrically connected to the bottom and top ITO electrodes, respectively, through micro-sized gold coated silica spacers. These spacers determine the thickness of the LC cavity and are held in place by UV cured optical glue. Alternatively, a hole can be etched in the GaAs substrate, inside which the VCSEL structure is grown and the remaining volume is filled in with the LC. A separate substrate, with a 20 nm thick gold layer deposited on it, is put upside down on top of the first substrate, after which the substrates are waferbonded [17]. This way an external LC cavity of thickness L, localized between the top VCSEL DBR and the top golden mirror, is formed. LC alignment layers would be normally put on both sides of the LC cavity - directly on the gold contact and on an additional 30 nm ITO layer on the VCSEL side. However, for simplicity of the analysis, we omit these layers and just assume that the LC molecules are initially oriented parallel to the y axis. This simplification does not influence the simulated tendencies and without it all L values would be reduced by the total thickness of the ITO and alignment layers (40-90 nm). The system consisting of DBR and golden mirror, which power reflectivities are 99.8% and 96%, leads to the emitted light intensity being 0.01% of the light intensity within active region. Structure details are listed in Table I. Every material used in our model has a refractive index consisting of n x, n y and n z = n x components, allowing us to introduce birefringence in the xoy plane. VCSEL diodes are slightly anisotropic by nature [21], [5], thus we introduce a small difference (0.0004) between n y and n x components for all epitaxial layers but the oxidation layer. As for the LC, we assume that it is an uniaxial nematic with the director (LC molecules aver-

FRASUNKIEWICZ et al.: POLARIZATION- AND MODAL-CONTROL IN A VERTICAL-CAVITY SURFACE-EMITTING LASER 5439 TABLE I LC-VCSEL PARAMETERS Material Refractive index Thickness [μm] along Ox: n x along Oy: n y Top metal mirror Au 0.2 i5.6 n x 0.02 E7 n x and n y according to (1) L n o = 1.51, n e = 1.71 LC VCSEL DBRs: Top/Bottom 24.5 / 29.5 pairs Al 0. 12 Ga 0. 88 As 3.52 + i10 4 n x +0.0004 0.0603 Al 0. 9 Ga 0. 1 As 3.11 + i10 4 n x +0.0004 0.0683 VCSEL oxide layer, aperture radius r ox Al 0. 9 Ga 0. 1 As 3.11 + i10 4 n x +0.0004 0.027 AlAs 2.95 i0.0 n x 0.015 AlOx 1.6 i0.0 n x Al 0. 9 Ga 0. 1 As 3.11 + i10 4 n x +0.0004 0.027 VCSEL 1λ cavity Al 0. 3 Ga 0. 7 As 3.11 + i10 4 n x +0.0004 0.1122 3 GaAs QW 3.62 + i0.0 n x +0.0004 0.024 Al 0. 3 Ga 0. 7 As 3.11 + i10 4 n x +0.0004 0.1122 aged long-axis orientation) rotating in the yoz plane and that it has an uniform distribution above the laser diode. Therefore, we can describe the LC state with a single angle θ between the molecules long-axis direction and the Oz axis (see the inset in Fig. 1). The refractive index of the LC is strictly connected to orientation and arrangement of the molecules. In general this refractive index has three orthogonal components n 1, n 2 and n 3 influencing light polarized along each of the crystalline axes. In an uniaxial nematic LC, the molecules are identical along the shortaxes with corresponding components n 1 = n 2 = n o, while for long-axis n 3 = n e, where n o and n e are, respectively, the ordinary and the extraordinary indices of refraction. For molecules oriented at an angle θ, as described above, we can obtain n x and n y components of the LC cell influencing light traveling along the Oz axis of two perpendicular polarizations. It is done by using index ellipsoid method [22], from which we obtain: n x = n o, n e n o n y = n 2 e cos 2 (θ)+n 2 o sin 2 (θ). (1) It is worth noting here, that more realistic approach, in which we consider varying θ(z) distribution in the LC cell above the laser, as well as slight molecule deviations, would produce results of the same tendencies. In such a case the impact of θ changes on the effective n y would be weaker [15]. III. SIMULATED TRANSVERSE MODES As we are utilizing a three-dimensional fully-vectorial optical model, we are able to analyze plethora of linearly-polarized transverse modes of the LC-VCSEL structure, including all of their degenerate forms (two for the fundamental and four for Fig. 2. (color online) Electrical field modal patterns in the xoy plane of the simulated VCSEL cavity for the first three low-order transverse modes confined within the oxide aperture (in rows: LP 01,LP 11 and LP 21 ) for all their degenerated forms (in columns). The red-blue (purple-green) color-scale indicate the electric field strength in the x (y) direction. higher-order modes), as shown in Fig. 2. We will limit our analysis to the first three transverse modes, namely LP 01,LP 11 and LP 21. Furthermore, in the case of higher-order modes, only one mode for each polarization will be considered, as two modes of the same polarization share the same tendencies in the simulations and would normally be degenerate due to the cylindrical symmetry of the device (e.g. x-lp 11a and x-lp 11b modes shown in Fig. 2 will be considered as a single x-lp 11 mode). Planewave decomposition is done using 722 plane-waves in each layer of the structure and the simulation-window is 1.5 r ox with perfectly matched layers (PML) located at the boundaries. Such simulation domain size was determined to provide most accurate results with reasonable computation time and without introducing boundary condition errors. We carry out a threshold analysis of the selected transverse modes, in which each of the resonant modes is investigated independently. In below threshold regime the driving currents are usually low, carrier distribution in the active region is mostly uniform, heat generation is limited and spatial hole burning effect is not present. For these reasons we can employ a cold-cavity approach, in which we consider only the optical phenomena. This is a commonly used simplification, as it significantly reduces the modeling complexity, while still giving accurate results [20]. From our calculations we obtain optical field distribution in the whole structure, wavelength and modal losses for each of the analyzed transverse modes. The modal losses of a mode are strictly related to its threshold gain and will be the parameter of choice to be used in our analysis. For any given structure configuration, the mode with the lowest modal losses is the mode to first reach its threshold, while the modal losses of the other modes are shown for assessing the mode discrimination strength. The single mode criterion is not precisely defined, however, single mode operation of a VCSEL can be assumed if SMSR is above 30 db, which occurs for very narrow aperture of oxide confined VCSEL (e.g. 3.5 μm) [23]. Since oxidation of such size results in limited output power

5440 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 23, DECEMBER 1, 2016 Fig. 3. (color online) Wavelength (top) and modal losses (bottom) change with LC thickness for the x-lp 01 (solid black line) and y-lp 01 (dashed green line) modes. LC birefringence is lifted by setting θ =0, aperture radius r ox = 3.5 μm. and quicker device aging, it is common to employ devices with wider aperture, working in a quasi single-mode regime of SMSR about 15 db [23]. In [20] we showed that for 980 nm VCSEL, SMSR close to 15 db corresponds to 9 cm 1 of modal losses difference between the fundamental mode and second one of lowest modal losses. Finding the threshold gains of the modes requires iterative calculations for fixed LC-VCSEL parameters while varying the material gain within the active region. We have performed such iterative calculations for representative cases, which indicated that the dependency between the modal losses (α) and threshold gain (g th ) is nearly linear and is given by formula: g th =87 α [cm 1 ]. IV. IMPACT OF LC-VCSEL GEOMETRY PARAMETERS ON MODE LASING CHARACTERISTICS A. LC Thickness 1) Periodic Switching of the LP 01 Orthogonally Polarized Modes: The influence of the LC thickness L on the fundamental modes of x and y linear polarization can be seen in Fig. 3. Here we set θ =0, thus removing the LC birefringence in the xoy plane, to show that in such a case, we obtain similar behavior of the modal losses of the two linearly polarized fundamental modes. The vertical shift visible between the wavelength curves of the two orthogonally polarized modes is due to the VCSEL anisotropy. There is also an unnoticeably small (in the scale of Fig. 3) horizontal shift between the modal losses characteristics, which will be investigated in detail later. 2) Loss-mechanism: Regions of significantly increased modal losses can be observed at regular intervals, starting from a very thin LC layer and continuing to L = 10 μm and beyond. The reason for the appearance of such regions is associated with a fact that the analyzed LC-VCSEL is a coupled-cavity structure [24] and is a consequence of tuning in-resonance and out-of-resonance of the two cavities [15]. It can be explained by looking at the on-optical-axis modal-profiles for different Fig. 4. (color online) Longitudinal optical power distributions for the two fundamental LC-VCSEL x-lp 01 modes: VCSEL mode (green) and LC mode (red) (see the text for description of these modes). Four different LC thicknesses L were chosen with regard to the tuning of the the two cavities - (a) right before resonance (L =5.03), (b) in-resonance (L =5.04), (c) right after resonance (L =5.05) and (d) away from the resonance (L =5.1). Black lines indicate the refractive index profile of the structure; the oxide-aperture radius is r ox = 3.5 μm. detunings between the VCSEL and the LC cavities, realized by changing the LC cavity thickness (Fig. 4). As typical for coupled-cavity system, two x-lp 01 modes exist one associated with the VCSEL cavity (most of the optical field in the VCSEL cavity) and one with the LC cavity (most of the optical field in the LC cavity). By looking at the case corresponding to a low-loss situation (Fig. 4(d)), it can be seen that the optical fields of these two compound cavity modes are strongly confined within their respective resonators. This is because in such a case the LC- and VCSEL-resonators are strongly detuned. By changing L, we periodically tune the two cavities in-resonance, causing the resonant compound modes of each cavity to partially transfer their optical field between the resonators, resulting in a significant part of the VCSEL cavity mode to reside in the LC cavity (Fig. 4(a) (c)). The net-losses are strictly connected to the amount of optical power confined within the gain region. By tuning the cavities in-resonance, the overlap between the mode-distribution and the gain region was found to be reduced by a factor of two. Within the in-resonance extremum, a significantly increased radiation leakage occurs and emitted light intensity increases from 0.01% to 1% of the intensity within the active region, which results in modal loss increase by about one order of magnitude. The regions of increased modal losses correspond to increase of the threshold current I th, accompanied by simultaneous increase in photon extraction efficiency. However, the relation between the modal losses and I th is not elaborated in this paper, as it is nonlinear in nature and would require a different simulation approach. 3) Multi Transverse-Mode Characteristics: With understanding of the cavity-tuning induced loss-mechanism, we can continue with investigation of the LC-VCSEL modal characteristics. A detailed insight into one of the interaction regions

FRASUNKIEWICZ et al.: POLARIZATION- AND MODAL-CONTROL IN A VERTICAL-CAVITY SURFACE-EMITTING LASER 5441 Fig. 5. (color online) Wavelength (top row) and modal-losses (bottom row) change with LC thickness for the first three low-order transverse modes of two orthogonal linear polarizations as indicated in the legend. Each column represents different oxide aperture radius: r ox = 2.5 μm (left), r ox = 3.5 μm (middle) and r ox = 4.5 μm (right). In this case, the LC director orientation is θ =0(the LC birefringence is lifted, and only the VCSEL birefringence is present). The mode with the lowest modal losses, for any given L, is the mode that will start lasing at the threshold. Vertical lines indicate modal switching points (where a change of the mode with the lowest losses occur) and the mode with the lowest losses is denoted by a letter at the bottom of each region between two lines: A x-lp 01, B y-lp 01,C x-lp 11,D y-lp 11,E x-lp 21,F y-lp 21. The lime-green regions indicate a loss difference of 9 cm 1, sufficient for a quasi single-mode operation for three different oxidation aperture radii, considering the first three low-order modes of both polarizations, is presented in Fig. 5. The vertical lines in this figure indicate modal switching and the dominant mode for each region between two lines (a mode with the lowest losses) is denoted by a letter at the bottom of each losses graph: A x-lp 01,B y-lp 01,C x-lp 11,D y-lp 11,E x-lp 21,F y-lp 21. It can be seen that the shorter the transverse-mode wavelength is, the thicker LC is required for that mode to experience in-resonance tuning. For this reason, with increasing L, the fundamental mode first come into interaction and is consequently followed by higherorder transverse-modes. This results in transverse-mode switching when reaching in-resonance tuning and fundamental mode dominance when going out of resonance region. The same principle applies to x- and y-polarizations of each transverse-mode. B. Aperture Radius Each column in Fig. 5 corresponds to a different oxidation aperture radius r ox. From the top row of Fig. 5, it is visible that an increase of r ox results in a red-shift for all the modes, where for the higher-order modes this effect is stronger. Simultaneously, the modal losses are reduced, this effect is again stronger for the higher-order modes. Such behavior is to be expected as it is typical for oxide-confined VCSELs and weakly index-guided waveguides in general [1, Ch. 4.3]. It is associated with the amount of modal optical-field leaking into the cladding region. The higher the mode-order, the wider its optical-field and stronger its overlap with the oxidation region is. Aside the aforementioned oxide-vcsel specific phenomena, a rather limited influence of the oxide aperture radius on the LC-VCSEL characteristics is observed. However, it can be noticed that the width of the disturbed regions increases with the mode-order. That effect is more visible in a case of smaller r ox. The reason for such behavior is that the LC-modes experience much weaker transverse-confinement than the VCSEL-modes. When the cavities become tuned in-resonance, the oxide aperture influence becomes significantly stronger for the LC-modes and weaker for the VCSEL-modes, extending the interaction region over a longer span of LC thicknesses. This leads to one side of the in-resonance modal losses slope to have increased modal discrimination. On example of r ox = 2.5 μm (Fig. 5(a)), the modal losses difference between LP 01 and LP 11 modes is 2.8 cm 1 for L = 4.95 μm (far from resonance) and 9.3 cm 1 for L = 5.08 μm (close to resonance). V. VOLTAGE CONTROLLED MODE-AND POLARIZATION-SWITCHING Electric field applied to the LC cell changes the LC-director orientation [22], [15]. Therefore voltage changes can be simulated by varying θ within 0 π/2 range, where θ = π/2 would correspond to a planarly-aligned LC case without voltage applied. Furthermore, n y determines the optical thickness of a LC cell for y-polarization, which is equal to n y (θ)l. Therefore it is possible to electro-optically modify the tuning between LC- and VCSEL-cavities for y-polarization, while keeping LC thickness and x-polarization characteristics constant. Several

5442 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 23, DECEMBER 1, 2016 Fig. 6. (color online) Electro-optical control of the LC-VCSEL modal and polarization properties, on example of the modal losses dependency on the director orientation θ, for the first three low-order transverse modes of two orthogonal linear polarizations at different LC thickness: L = 5.075 μm (a),l = 5.04 μm (b) and L = 5.02 μm (c). Oxide-aperture radius is r ox = 3.5 μm. As in Fig. 5, the vertical lines indicate modal switching and the dominant mode for each region between two lines (a mode with the lowest losses) is denoted by a letter at the bottom of each graph: A x-lp 01,B y-lp 01,C x-lp 11,D y-lp 11,E x-lp 21,F y-lp 21. In (c) the 3 lines corresponding to x-polarized modes overlap. The light-yellow regions indicate enhanced modal discrimination with respect to the fundamental mode of x polarization. representative LC thicknesses L were chosen, based on the tendencies presented in Fig. 5, to show the possibility of achieving different modal- and polarization-switching behaviors. Calculations presented in this section were performed for r ox = 3.5 μm. A. Fundamental Mode Polarization Switching and Increased Discrimination of Higher-Order Modes In order to achieve polarization control of the fundamental mode, while avoiding lasing of higher-order transverse modes, a LC thickness slightly bigger than the one satisfying the exact in-resonance condition for the LP 01 modes should be chosen (Fig. 6(c)). In such a case a well defined regions of x or y fundamental mode emission can be obtained. Furthermore, for θ favoring x-lp 01 mode, a significantly increased discrimination of all other higher-order modes is achieved, e.g. the losses of the x-lp 11 first-order mode are 60% higher than the ones for the x-lp 01 mode. In contrast, analogous discrimination in the case of a dominant y-polarization is about 15%. By biasing the LC-VCSEL with a current compensating for modal losses of 13 cm 1, a change of LC voltage would cause switching between y-polarized strongly multimode and x-polarized nearly single mode operation. B. Polarization Discrimination and Transverse Mode Switching When a strong discrimination of one polarization is desired, a LC thickness corresponding to an in-resonance condition should be chosen, granting constant high modal-losses for the x-polarization modes (Fig. 6(b)). In such a case the y-lp 10 will be dominant over the whole θ range. Additionally, by bringing the y-polarization cavity modes into a resonance with increasing θ (the left side of the resonance maxima), consecutive switches to the y-lp 11 and y-lp 21 modes can be observed. Finally, in the middle of the resonance region all modes of both polarizations are experiencing high losses. C. Increased Laser Instability By looking at Fig. 5 it can be seen, that when the LC and VCSEL cavities are tuned in-resonance by changing the LC thickness L, a region of strongly reduced modal-discrimination is present on the left side of the losses maxima. The modal losses of all the (x-polarized) modes are reaching the same value and appear to cross at one point. For such an L (Fig. 6(a)), analogous procedure can be then applied to the y-polarization modes through a proper θ selection. High modal and polarization instabilities of the LC-VCSEL are expected for such conditions, see e.g. [5]. VI. CONCLUSION We have related the modal and polarization selectivity mechanisms in a VCSEL with liquid crystal (LC) overlay to the tuning between the coupled VCSEL and LC cavities. We have shown how the modal characteristics of such an LC-VCSEL are influenced by the LC thickness, oxidation aperture radius and electro-optically tuned LC director orientation. Finally, we have

FRASUNKIEWICZ et al.: POLARIZATION- AND MODAL-CONTROL IN A VERTICAL-CAVITY SURFACE-EMITTING LASER 5443 identified parameter regions allowing for a voltage-controlled polarization switching of the fundamental LP 01 modes, or switching to higher-order transverse modes with the same polarization. Quite importantly, we have demonstrated the possibility of modifying higher transverse-order mode discrimination, which could be either increased for enhanced single mode emission, or decreased resulting in modal and/or polarization instabilities. It could be noted, that quite high precision of determining the LC thickness (about 10 nm) is required to achieve different operation regimes. Such manufacturing accuracy would be difficult to achieve, however, LC have relatively high thermo-optic coefficient ( n/ T of about 10 3 K 1 and higher), thus precise tuning of the LC cavity optical-thickness can be achieved by means of ambient temperature (T ) control. The strength of this effect is proportional to the LC thickness, as the thicker the LC layer is, the smaller the refractive index change is required to produce a certain absolute difference in optical-thickness. Assuming n/ T =10 3 K 1, LC optical-thickness change of Δ(L n) =10nm would require ambient temperature change of ΔT =1.5KforL = 10 μm and ΔT =6KforL = 2.5 μm. Simultaneously, the VCSEL cavity will also be thermally affected, however, the VCSEL resonator is only 0.25 μm thick and AlGaAs thermo-optic coefficient is of about 10 4 K 1. Therefore, the VCSEL cavity influence is negligible in comparison to the LC cavity. Finally, we would like to stress that although our analysis has been carried out for a GaAs VCSEL lasing around 850 nm, the presented tendencies are generally valid for VCSELs based on different material systems as the physical mechanism of electro-optical tuning of coupled VCSEL and nematic liquid crystal cavities remains the same. REFERENCES [1] R. Michalzik, VCSELs Fundamentals, Technology and Applications of Vertical-Cavity Surface-Emitting Lasers. Berlin, Germany: Springer- Verlag, 2013. [2] M. Shimizu, F. Koyama, and K. Iga, Polarization characteristics of MOCVD grown GaAs/GaAlAs CBH surface emitting lasers, Jpn. J. Appl. Phys., vol. 27, no. 9, pp. 1774 1775, 1988. [3] K. D. Choquette, R. P. Schneider, Jr K. L.Lear, and R. E. Leibenguth, Gain dependent polarization properties of vertical cavity lasers, IEEE J. Sel. Topics Quantum Electron., vol. 1, no. 2, pp. 661 666, Jun. 1995. [4] C. J. Chang-Hasnain, J. P. Harbison, L. T. Florez, and N. G. Stoffel, Polarization characteristics of quantum well vertical cavity surface emitting lasers, Electron. Lett., vol. 27, no. 2, pp. 163 165, 1991. [5] K. Panajotov et al., Polarization behavior of vertical-cavity surfaceemitting lasers: Experiments, models and applications, in Nanoscale Linear and Nonlinear Optics, M. Bertolotti, C. M. Bowden, and C. Sibilia, Eds. vol. 560, 2001, pp. 403 417. [6] P. Debernardi and G. P. Bava, Coupled mode theory: A powerful tool for analyzing complex VCSELs and designing advanced device features, IEEE J. Sel. Topics Quantum Electron., vol. 9, no. 3, pp. 905 917, May/Jun. 2003. [7] M. Grabherr et al., Volume production of polarization controlled singlemode VCSELs, Proc. SPIE, vol. 6908, 2008, Art. no. 690 803. [8] M. Sciamanna and K. Panajotov, Two-mode injection locking in verticalcavity surface-emitting lasers, Opt. Lett., vol. 30, pp. 2903 2905, 2005. [9] I. Gatare, M. Sciamanna, J. Buesa, H. Thienpont, and K. Panajotov, Nonlinear dynamics accompanying polarization switching in vertical-cavity surface-emitting lasers with orthogonal optical injection, Appl. Phys. Lett., vol. 88, 2006, Art. no. 101106. [10] I. Gatare, J. Buesa, H. Thienpont, K. Panajotov, and M. Sciamanna, Polarization switching bistability and dynamics in vertical-cavity surfaceemitting laser under orthogonal optical injection, Opt. Quantum Electron., vol. 38, pp. 429 443, 2006. [11] K. Panajotov, I. Gatare, A. Valle, H. Thienpont, and M. Sciamanna, Polarization- and transverse-mode dynamics in optically injected and gain-switched vertical-cavity surface-emitting lasers emitting lasers, IEEE J. Quantum Electron., vol. 45, no. 11, pp. 1473 1481, Nov. 2009. [12] C. Masoller and M. S. Torre, Influence of optical feedback on the polarization switching of vertical-cavity surface-emitting lasers, IEEE J. Quantum Electron., vol. 41, no. 4, pp. 483 489, Apr. 2005. [13] K. Panajotov, M. Sciamanna, M. Arizaleta, and H. Thienpont, Optical feedback in vertical-cavity surface-emitting lasers, IEEE J. Sel. Topics Quantum Electron., vol. 19, Jul./Aug. 2013, Art. no. 1700312. [14] C. I. Wilkinson, J. Woodhead, J. E. F. Frost, J. S. Roberts, R. Wilson, and M. F. Lewis, Electrical polarization control of vertical-cavity surfaceemitting lasers using polarized feedback and a liquid crystal, IEEE Photon. Technol. Lett., vol. 11, no. 2, pp. 155 157, Feb. 1999. [15] K. Panajotov and H. Thienpont, Vertical-cavity surface-emitting laser with liquid crystal overlay, Opt. Express, vol. 19, no. 18, pp. 16 749 16 759, 2011. [16] Y. Xie, J. Beeckman, K. Panajotov, and K. Neyts, Vertical-cavity surfaceemitting laser with a liquid crystal external cavity, Opt. Lett., vol. 39, pp. 6494 6497, 2014. [17] Q. Wang, B. Guan, G. Shi, X. Liu, X. Guo, and G. Shen, Wavelength tunable VCSELs based on voltage-dependent birefringence of liquid crystal, Proc. SPIE, vol. 8552, 2012, Art. no. 85 520S. [18] M. Dems, R. Kotynski, and K. Panajotov, Plane wave admittance method, a novel approach for determining the electromagnetic modes in photonic structures, Opt. Express, vol. 13, pp. 3196 3207, 2005. [19] T. Czyszanowski, M. Dems, and K. Panajotov, Single mode condition and modes discrimination in photonic-crystal 1.3 mu m AlInGaAs/InP VCSEL, Opt. Express, vol. 15, no. 9, pp. 5604 5609, 2007. [20] T. Czyszanowski, M. Dems, R. P. Sarzała, K. Panajotov, and K. D. Choquette, Photonic crystal VCSELs: Detailed comparison of experimental and theoretical spectral characteristics, IEEE J. Sel. Topics Quantum Electron., vol. 19, no. 5, Sep./Oct. 2013, Art. no. 1701908. [21] A. K. Jansen van Doorn, M. P. van Exter, and J. P. Woerdman, Effects of transverse anisotropy on VCSEL spectra, Electron. Lett., vol. 30, no. 23, pp. 1941 1943, 1994. [22] I.-C. Khoo, Nematic liquid crystals, in Liquid Crystals. Hoboken, NJ, USA: Wiley, 2007, ch. 3. [23] P. Moser et al., Impact of the aperture diameter on the energy efficiency of oxide-confined 850 nm high speed VCSELs, Proc. SPIE, vol. 8639, 2013, Art. no. 86390V. [24] R. P. Stanley, R. Houdré, U. Oesterle, M. Ilegems, and C. Weisbuch, Coupled semiconductor microcavities, Appl. Phys. Lett., vol. 65, pp. 2093 2095, 1994. Authors biographies not available at the time of publication.